Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus

It is proved that if a normal semifinite weight $\varphi $ on a von Neumann algebra $\mathscr M$ satisfies the inequality $\varphi (|a_1+a_2|)\le \varphi (|a_1|)+\varphi (|a_2|)$ for any selfadjoint operators $a_1,a_2$ in $\mathscr M$ , then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality $|\varphi (a)|\le \varphi (|a|)$ is refined and reinforced.